Re: Replacement of Cardinality (infinite middle)

On 09/06/2024 12:40 PM, Jim Burns wrote:

On 9/5/2024 4:14 PM, Ross Finlayson wrote:

On 09/05/2024 12:57 PM, Ross Finlayson wrote:

On 09/03/2024 01:50 PM, Jim Burns wrote:

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Back in the 80's and 90's
it was Nelson's Internal Set Theory
where it was figured that
the avenue toward true non-standard real analysis
was to result.

>
This "true non-standard real analysis" must concern
something other than
the Dedekind.complete ordered field.
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I.e.,
not-a-real-functions with real analytical character,
like Dirac's delta function or
here for example
the Natural/Unit Equivalency Function,
it is expected that
"foundations" _does_ formalize them, and that
what doesn't, simply, isn't,
respectively.

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You (RF) may be tired of nuance by now,
but
I think we need to distinguish between
what _simply_ isn't and
what _a specific foundation_ won't say is.
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Consider Boolos's ST as a toy foundation.
⎛ ∃{}
⎜ ∃z = x∪{y}
⎝ extensionality
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ST supports the existence of each finite ordinal
via a finite not.first.false claim.sequence.
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ST does not support the existence of
a set of all finite ordinals.
At least, I don't see how it could.
ST doesn't support its non.existence, either.
At least, I don't see how it could.
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An ordinal which has itself as an element
simply isn't.
That depends pretty much completely on
_what ordinal are_  well.ordered.
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Getting around that prohibition would
require ordinals which were something else.
But that's not actually getting around it.
That's only playing a game similar to
"if we rename 2 as 3, then 1+1=3"
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Then this "infinite middle" is just about
the simplest "non-Archimedean" that there is,
and in fact even simpler, than for example
axiomatizing "0" and "omega"

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"omega" must be
something other than
the first transfinite ordinal.
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axiomatizing "0" and "omega"
with an infinite-middle pretty much
exactly like ZF does,
except symmmetric about the middle
instead of non-inductive yet declared fiat
(stipulated).

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1+1=3?
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Oh?  "Ken:  2 + 2 = 4".
ZF introduces two constants, 0 for the empty set,
and omega w for the first countably-infinite and
limit ordinal, countably-infinite in a usual model of
ordinals in set theory like v.N.'s where order is modeled
as by transitive containment, where of course the relation
in set theory is "elt" not "contains", and the definition of
ordinal is sort of reverse or _after_ otherwise comprehension.
(That most people ignore as confused, conflated, confounded,
vis-a-vis confounding, conflating, ..., and confusing, meaning
for the roots of the words not any matters of actual uncertainty
or lack of definition, that instead con-foundation and con-flation
and con-fusion are reflections of impredicativity after intended
or naive ellisions of otherwise contradistinction. That most
people ignore, ..., the difference between "elt" and "contains",
and since what.)
A _foundation_, then, is a theory underpinning and provides
a grounds or alaya a fundament as of a firmament, typically
the analytical and elementarily fundamental in the usual
dialectic of axiomatics the independence of axioms as with
regards to their various _expansion_ of comprehension, then
as with regards to what are various _restrictions_ of comprehension,
which somehow "the foundation", proper and singular, does
not have.
So, some have the axiomatization as introduction as constants
of "empty set" and "infinite set" as _expansion_ of comprehension,
while others have them in at least in part _restriction_ of comprehension.
Then, a wider theory makes what are otherwise "uniqueness" results
into "distinctness" results, more that i^4 = (-1)^2 = 1 than (nonsense
omitted).
Then, modeling modularity for example for integers as smallest
and greatest and infinite in the middle is quite well-defined
and very well subject to comprehension.
Or, your nuances are not lost, and in fact, found.